Optimal. Leaf size=14 \[ -\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 65, 213}
\begin {gather*} -\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1+x^8}} \, dx &=\frac {1}{8} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^8\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^8}\right )\\ &=-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.95, size = 6, normalized size = 0.43 \begin {gather*} -\frac {\text {ArcSinh}\left [\frac {1}{x^4}\right ]}{4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 19, normalized size = 1.36
method | result | size |
trager | \(\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{x^{4}}\right )}{4}\) | \(17\) |
default | \(\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{\sqrt {x^{8}}}\right )}{4}\) | \(19\) |
meijerg | \(\frac {\left (-2 \ln \left (2\right )+8 \ln \left (x \right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{8}+1}}{2}\right )}{8 \sqrt {\pi }}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (10) = 20\).
time = 0.26, size = 25, normalized size = 1.79 \begin {gather*} -\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (10) = 20\).
time = 0.33, size = 25, normalized size = 1.79 \begin {gather*} -\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 8, normalized size = 0.57 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{4}} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (10) = 20\).
time = 0.00, size = 35, normalized size = 2.50 \begin {gather*} \frac {\frac {\ln \left (\sqrt {x^{8}+1}-1\right )}{2}-\frac {\ln \left (\sqrt {x^{8}+1}+1\right )}{2}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 10, normalized size = 0.71 \begin {gather*} -\frac {\mathrm {atanh}\left (\sqrt {x^8+1}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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